Obections to Cantor's Theory (Wikipedia article)
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From: Chris Menzel on 25 Jul 2005 11:54 On Thu, 21 Jul 2005 12:48:54 +0200, Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> said: > Chris Menzel wrote: >> Guess what? That means you reject the continuum hypothesis! Isn't >> that exciting? Don't you just want to go out and learn all about it >> rather than just spouting vague, uninformed, and often silly >> nonsense? > > Really, do you mean that _accepting_ the Continuum Hypothesis somehow > represents no-nonsense behaviour ? Really, do you mean that you think my comment above has any such implication? TO, in the comment to which I was responding, essentially asserted that he believed CH, though with typical cluelessness he was unable to recognize the fact. I was simply it pointing out to him, and suggesting he might want actually to study the mathematics instead of continuing to waste his time making an utter fool of himself. Chris Menzel
From: Tony Orlow on 25 Jul 2005 12:09 Robert Low said: > Daryl McCullough wrote: > > That's not true. If S is an infinite set of strings, then there > > is a difference between (1) There is no finite bound on > > the lengths of strings in S. (2) There is a string in S that is > > infinite. > > Except that TO claims that (1) implies (2), though I can't > even get far enough into his head to see why he thinks it, > never mind finding his 'argument' convincing. > > Funnily enough, there is a similar sounding statement that > is true in non-standard analysis: any set containing arbitrarily > large finite integers must also contain an infinite integer. > But in that game, the class of all finite integers isn't > a set :-) > > I only mentioned this because I thought it might muddy the > waters in an entertaining way... > Sure, that sounds like good mud. The set of all finite integers is a poorly defined set, an "indeetrminate" set, with no clear boundary. I can see, intuitively, that a set that contains arbitrarily large finite values must include an infinite value, although i am not sure what their proof relies on. Now, what do you not understand about N=S^L. The number of binary strings of length L is 2^L, so you cannot have an infinite set of binary strings unless L is allowed to be infinite, in which case the binary value is infinite. There can only be a finite number of finite strings with a finite alphabet. -- Smiles, Tony
From: Daryl McCullough on 25 Jul 2005 12:11 Tony Orlow (aeo6) wrote: >The proof regarding strings is so simple, you really can't complain. N=S^L >precisely describes the relationship between the number of strings N, the >symbol set size S, and the length of strings L. But for the set we are talking about, there *is* no L. We're talking about the set of *all* finite strings. That's an infinite union: If A_n = the set of all strings of length n, then the set of all possible finite strings is the set A = union of all A_n = { s | for some natural number n, s is in A_n } This set has strings of all possible lengths. So there is no L such that size(A) = S^L. >I am not assuming anything except for this fact. You are assuming that every set of strings has a natural number L such that every string has length L or less. That's false. -- Daryl McCullough Ithaca, NY
From: Chris Menzel on 25 Jul 2005 12:10 On Thu, 21 Jul 2005 09:57:21 +0200, Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> said: > Virgil wrote: > >> Every bit of "Cantorianism" has been well enough defined for the >> understanding of thousands upon thousands of people. That TO fails where >> so many have succeeded says more about TO than about the adequacy of >> "Cantorianism's" explanations. > > The fact that a faith has millions of adherants doesn't say anything > about its validity. It says something about the society wherein it is > accepted, though. Faith?? "Cantorianism" is embodied in a completely rigorous axiomatic theory. The propositions you find unacceptable are demonstrably valid in that theory. There is not a lick of faith involved. Instead of tossing off idiotic comparisons to religious belief -- an inevitable rhetorical haven for cranks and crackpots -- you might consider a genuinely mathematical response: point out the axiom(s) of set theory you consider unacceptable and defend your rejection of them with arguments; or simply embark straightaway on the development of an equally rigorous alternative. Responses like yours only show you haven't the least clue what mathematics is. Chris Menzel
From: imaginatorium on 25 Jul 2005 12:56
Tony Orlow (aeo6) wrote: > I don't see where you pointed out any specific flaw, except to rant about your > largest finite number again. No, well, I give up. Just for my curiosity, though, I still cannot understand your point when you complain about "ranting about my[sic] largest finite number". It has been pointed out to you so many times - with absolutely no effect - that the Peano axioms (or any similar more informal notion of pofnats) imply that there cannot be a largest pofnat. Just tell me: do you claim... (1) There _is_ a largest pofnat. (2) There is no largest pofnat (but the contradictions with your ideas escape you) (3) The answer to "Is there a largest pofnat?" is somehow neither 'Yes' nor 'No'. Thanks. Brian Chandler http://imaginatorium.org |